Since we are trying to get an equation with only x's and y's, let's look back at the relationships we have:

x = cos(theta)
y = sin(theta)

I can square each equation to get:

x^2 = cos^2(theta)
y^2 = sin^2(theta)

But recall that cos^2(theta) = 1 - sin^2(theta), so the x^2 equation becomes:

x^2 = 1 - sin^2(theta)

And we know that sin^2(theta) = y^2, so we get:

x^2 = 1 - y^2
x^2 + y^2 = 1

Notice that to get the "x" equation in terms of only x and y we needed to change the cos(theta) into a sin(theta) so that we could replace the sin(theta) with y. The only way to do this is to square both sides of each equation.